Combinations - With Restriction


Suppose 55 distinct numbers are chosen from nine integers 1,2,,91, 2, \ldots, 9 to create a 55-digit number. Given that the digits chosen must consist of 33 odd numbers and 22 even numbers, how many distinct 55-digit numbers can be created?

3 red balls and 1212 blue balls are randomly placed in a row. What is the probability that no two red balls are adjacent?

Suppose a 44-digit number is created from 44 distinct integers chosen among {1,2,3,4,5,7,9}.\{1, 2, 3, 4, 5, 7, 9\}. How many such 44-digit numbers include both 11 and 22 but neither begin nor end with 1?1?

In Sue's Fruit & Flowers Market, there are 88 different kinds of fruit, including peaches and apples. There are also 66 different kinds of flowers, including roses and lilies. How many ways are there to make a gift basket such that the basket has 33 different kinds of fruit including peaches and 22 different kinds of flowers with no roses?

From a group of 1111 students including AA and BB, 44 people are selected to be arranged in a line. If both AA and BB must be among the 44 selected students, how many distinct line-ups are possible?


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